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Math Help - Egyptian Fractions

  1. #1
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    Egyptian Fractions

    Challenge Question:

    Wiki says that Egyptian Fractions are distinct unit fractions so e.g. 1/2, 1/3, 1/5 are Egyptian Fractions while 2/4 or 3/9 are not; also you can't repeat the fraction.

    Fractions that aren't Egyptian Fractions can be expressed as Egyptian Fractions so, for example, you can have 5/7 = 1/3 + 1/4 + 1/8 + 1/168.

    Here's your challenge. Can you express 5/7 as 1/x + 1/y + 1/z where x,y and z are distinct positive integers. If you can't do it, then I'll give the answer myself in about a week.

    Have fun.



    Moderator edit: This is an approved challenge question.
    Last edited by mr fantastic; April 3rd 2010 at 03:41 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by wonderboy1953 View Post
    wiki says that egyptian fractions are distinct unit fractions so e.g. 1/2, 1/3, 1/5 are egyptian fractions while 2/4 or 3/9 are not; also you can't repeat the fraction.

    Fractions that aren't egyptian fractions can be expressed as egyptian fractions so, for example, you can have 5/7 = 1/3 + 1/4 + 1/8 + 1/168.

    Here's your challenge. Can you express 5/7 as 1/x + 1/y + 1/z where x,y and z are distinct positive integers. If you can't do it, then i'll give the answer myself in about a week.

    Have fun.
    Seems a bit easy:

    Spoiler:

    5/7=1/2+1/7+1/14
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  3. #3
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    Excellent CB

    This problem is inspired by Tony Crilly's book 50 Mathematical Idea. He indicates that it's still unknown how far you can condense Egyptian fractions which people may wish to explore. I'm also wondering whether any (rational) fraction can be expressed as Egyptian Fractions.
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  4. #4
    Super Member Deadstar's Avatar
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    Spoiler:


    Ok, my method was mildly brute force, but with a bit of logic involved...

    Not that 1/4 + 1/5 + 1/6 is the largest (in value after addition) fraction sequence we can have that begins with 1/4 and this is < 5/7

    Hence we have reduced this to two cases, a sequence beginning with 1/x = 1/2 or 1/3.

    Taking 1/x = 1/3 first and again applying the above idea, we see that we must now have 1/y = 1/4 as 1/5 + 1/6 is less than 5/7 - 1/3 = 8/21.

    So having 1/y = 1/4 gives 1/z = 11/84, hence we must take 1/x =1/2.

    Doing this results in 5/7 - 1/2 = 1/y + 1/z.

    => 3/14 = 1/y + 1/z.

    Now clearly we can take 1/y = 2/14 = 1/7 and 1/z = 1/14.

    5/7 = 1/2 + 1/7 + 1/14.
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by wonderboy1953 View Post
    This problem is inspired by Tony Crilly's book 50 Mathematical Idea. He indicates that it's still unknown how far you can condense Egyptian fractions which people may wish to explore. I'm also wondering whether any (rational) fraction can be expressed as Egyptian Fractions.
    "Every positive rational number can be represented by an Egyptian fraction." - Wikipedia
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  6. #6
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    The solution given above is not the only one!

    Spoiler:
    \frac57 = \frac12 + \frac15 +\frac1{70} = \frac12 + \frac16 + \frac1{21}
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  7. #7
    Super Member Deadstar's Avatar
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    Quote Originally Posted by Opalg View Post
    The solution given above is not the only one!

    Spoiler:
    \frac57 = \frac12 + \frac15 +\frac1{70} = \frac12 + \frac16 + \frac1{21}
    Just out of interest how did you compute yours? When I got to my last step I saw my answer immediately but I also realized there could potentially be solutions for...

    Spoiler:

    y = 5,6,7,8


    Was it just a case of checking through them one by one? Is there a more elegant solution to finding the values?
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by chiph588@ View Post
    "Every positive rational number can be represented by an Egyptian fraction." - Wikipedia
    I don't recall the details of the proof at the moment but the greedy algorithm always returns a EF decomposition for any rational.

    CB
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  9. #9
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    A deeper question

    Can any number (meaning a non-transcendental irrational number or a transcendental number) be expressed by a neverending series of Egyptian fractions? How would they compare with other series including continuing fractions?
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  10. #10
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    Quote Originally Posted by wonderboy1953 View Post
    Can any number (meaning a non-transcendental irrational number or a transcendental number) be expressed by a neverending series of Egyptian fractions? How would they compare with other series including continuing fractions?
    Consider the binary representation of a real. This can be rewritten as a possibly non-terminating EF. Also given a real the greedy algorithm can be applied and it will produce a convergent Egyptian series.

    CB
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